Hostname: page-component-7dd5485656-wxk4p Total loading time: 0 Render date: 2025-10-31T22:22:46.753Z Has data issue: false hasContentIssue false
  • English
  • Français

On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals

Published online by Cambridge University Press:  14 July 2016

E. K. Kyprianou
E. K. Kyprianou*
Affiliation:
University of Manchester
Get access Rights & Permissions [Opens in a new window]

Extract

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt, and the service time has distribution dB(t), 0 < t < ∞. Let W(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable denote the first busy period initiated by a waiting time u at time t = 0.

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 8 , Issue 3 , September 1971 , pp. 494 - 507
Copyright
Copyright © Applied Probability Trust 1971 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Benes, V. E. (1957) On queues with Poisson arrivals. Ann. Math. Statist. 28, 670–677.CrossRefGoogle Scholar
[2] Daley, D. J. (1969) Quasi-stationary behaviour of left-continuous random walk. Ann. Math. Statist. 40, 532–539.CrossRefGoogle Scholar
[3] Darroch, J. N. and Seneta, E. (1965) On quasi-stationary distributions in absorbing discrete time finite Markov chains. J. Appl. Prob. 2, 88–100.CrossRefGoogle Scholar
[4] Darroch, J. N. and Seneta, E. (1967) On quasi-stationary distributions in absorbing continuous time finite Markov chains. J. Appl. Prob. 4, 192–196.CrossRefGoogle Scholar
[5] Doetsch, E. (1958) Einführung in Theorie und Anwendung der Laplace-Transformation. Birkhauser Verlag, Basel.CrossRefGoogle Scholar
[6] Hille, E. (1959) Analytic Function Theory. Vol. I. Blaisdell.Google Scholar
[7] Mandl, P. (1959) Sur le comportement asymptotique des probabilités dans les ensembls des états d'une chaîne de Markov homogène. Casopis. Pest. Mat. 84, 140–149 (In Russian).CrossRefGoogle Scholar
[8] Mandl, P. (1960) On the asymptotic behaviour of the probabilities within groups of states of a homogeneous Markov process. Casopis. Pest. Mat. 85, 448–456 (In Russian).CrossRefGoogle Scholar
[9] Prabhu, N. U. (1965) Queues and Inventories. John Wiley & Sons Inc., New York.Google Scholar
[10] Seneta, E. and Vere-Jones, D. (1967) On quasi-stationary distributions in discrete time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434.CrossRefGoogle Scholar
[11] Takács, L. (1955) Investigation of waiting time problems by reduction to Markov processes. Acta Math. 6, 101–129.Google Scholar
[12] Vere-Jones, D. (1969) Some limit theorems for evanescent processes. Aust. J. Statist. 11, 67–78.CrossRefGoogle Scholar